Chapter 17, Problem 8RE

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Solve the differential equation.8. d 2 y d x 2 + 4 y = sin 2 x

To determine

To solve: The differential equation.

Explanation

Given data:

The differential equation is,

d2ydx2+4y=sin2x

yâ€³+4y=sin2x (1)

Consider the auxiliary equation.

r2+4=0 (2)

Roots of equation (2) are,

r=âˆ’0Â±(0)2âˆ’4(1)(4)2(1)â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰{âˆµr=âˆ’bÂ±b2âˆ’4ac2aforâ€‰theâ€‰equationâ€‰ofar2+br+c=0â€‰â€‰}=Â±i42=Â±2i

Write the expression for the complementary solution of two complex roots r=Î±Â±iÎ² .

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x) (3)

Substitute 0 for Î± and 2 for Î² in equation (3),

yc(x)=e0x(c1cos2x+c2sin2x)

yc(x)=c1cos2x+c2sin2x (4)

The Right hand side (RHS) of a differential equation contains only sine function, Therefore, the trail solution yp(x) for this case can be expressed as follows.

yp(x)=Axcos2x+Bxsin2x (5)

Differentiate equation (5) with respect to x.

yâ€²p(x)=ddx(Axcos2x+Bxsin2x)

yâ€²p(x)=(A+2Bx)cos2x+(Bâˆ’2Ax)sin2x (6)

Differentiate equation (6) with respect to x

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started